The resolution of the universal ring for finite length modules of projective dimension two
| dc.creator | Kustin, Andrew R. | |
| dc.date | 2006-07-25 | |
| dc.date.accessioned | 2026-07-07T07:20:55Z | |
| dc.date.available | 2026-07-07T07:20:55Z | |
| dc.description | Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0\to \Cal R^{e}\to \Cal R^{f}\to \Cal R^{g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\Cal R\to S$ with $\Bbb V=\Bbb U\otimes_{\Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ by free $\pmb K\otimes_{\Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{\Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2\times 2$ minors of an $e\times g$ matrix. | |
| dc.description | See also http://www.math.sc.edu/~kustin | |
| dc.identifier | https://arxiv.org/abs/math/0607639 | |
| dc.identifier | http://arxiv.org/abs/math/0607639 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115122 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13D25 | |
| dc.title | The resolution of the universal ring for finite length modules of projective dimension two | |
| dc.type | text |